Multicriticality and quantum fluctuation in a generalized Dicke model

Quantum many-body systems that support high-order quantum phase transitions are quite rare. Here we consider an important generalization of the Dicke model in which an ensemble of multilevel atoms, instead of two-level atoms as in the conventional Dicke model, interact with a single photonic mode, and show that this generalized Dicke model can become multicritical. For a subclass of experimentally realizable schemes, multicritical conditions of arbitrary order can be expressed analytically in compact forms. As such, experiments can be readily designed to achieve quantum phase transition of desired order. We also calculate the atom-photon entanglement entropy for both critical and noncritical cases and find that the order of the criticality strongly affects the critical entanglement entropy: higher order yields stronger entanglement. Our work provides deep insight into quantum phase transitions and multicriticality.

Figure 1

The experimental scheme for realizing a T-class multicritical Dicke model. Using a pair of Raman beams with opposite circular polarization, as marked by the dashed and the dotted lines, respectively, and a linearly $\pi$-polarized cavity mode, as marked by the solid line, it is possible to couple adjacent hyperfine states of ${}^{85}\mathrm{Rb}$ atoms through Raman coupling. By proper choice of the laser parameters, the coupling between “next-nearest” states (i.e., states with $\mathrm{\Delta }{m}_F=\pm 2$) through two Raman beams can be made to vanish because of the sum rule of the Clebsch-Gordan coefficients.

Figure 2

(a, b) The mean-field phase diagram of a four-level T-class Hamiltonian around the fourth-order critical point. The color bar represents the order parameter $\varphi$. The dashed line in (a) is an ordinary critical line while the solid line in (b) is a tricritical line. (c) The boundary that separates the normal phase (above) from the superradiant phase (below). The $h_{22}=2$ plane is the critical manifold, with a tricritical line marked by a solid line. The curved surface is the first-order phase transition boundary. In all panels, we use a white dot to mark the fourth-order critical point.

Figure 3

(a) The gap $\mathrm{\Delta }$ between the ground state and the first excited state, and (b) the ground-state entropy $S$, of the two-level Dicke model with $d_{12}=\sqrt{2}$ for different atom number $N$. $\mathrm{\Delta }$ exponentially decreases when $h_{22}$ moves deep into the superradiant phase. (c) Critical atom-photon entanglement entropy $S_{\mathrm{cri}}$ plotted against the atom number $N$ for T-class Hamiltonians with different orders of criticality. Different orders of criticality are achieved by first setting $(h_{33},h_{44},h_{55})=(3,3,2)$, and then tuning some $h_{kk}$ to very large values. For example, to achieve the tricriticality, we set $h_{44}$ and $h_{55}$ to be large.

Figure 4

(a) The phase diagram associated with a free energy $F$ that is a cubic polynomial in ${\varphi }^2$. The solid lines are the phase boundaries. The region to the left of the boundary marked by A and C is the symmetry-breaking phase and the right marked by B and D is the symmetry-preserving phase. The red solid line in the region $c_2>0$ is the critical line. The green solid line in the region $c_2<0$ is the triple line where three phases coexist. Regions C and D within the blue dashed line have three local minimums in the free energy. The black dot in the center is the tricritical point. (b) Typical plots of $F$ against $\varphi$ for different regions shown in (a).